Many basic mathematical ideas historically arose from real life problems. But are all basic mathematical ideas invented to serve a real-life purpose? Aarnout Brombacher, a South African consultant on mathematics education, advocates motivating students by posing problems, but points out that not every problem can be rooted in our experiences in an authentic way. For example, the way we multiply negative numbers is a result of extending certain properties of positive numbers, such as the distributive property, to apply to negative numbers. But unlike multiplication of positive numbers, which can be thought of finding area, there is no concrete representation of multiplication of negatives.

Brombacher's column is the eighth in a series discussing elementary mathematics education. In a previous column Brombacher suggested that teachers preempt the question "Why are we doing this?" by simply starting the lesson with a real-life problem. But in this column, he admonishes teachers to avoid concocting a "real life" problem to introduce an abstract concept since it will no doubt be awkward and unsatisfying to students. This begs the question: how can teachers motivate students to think abstractly? If a teacher should discuss abstract concepts using only abstract mathematical reasoning, they must first build bridges between concrete examples and abstract thinking.

Without addressing how this bridge is constructed, Brombacher also provokes the reader to think about the various historical leaps made in mathematics: from rational to irrational, from real to imaginary. Were these leaps made because of an advance in abstract thinking or because of a demand to solve a real-life problem?

University of Manitoba math professor Gabor Lukacs is seeking an injunction against his university in regard to a PhD degree that has already been awarded to a math graduate student. A letter from 86 mathematicians from nine countries, including Canada, supports his case. Lukacs is currently suspended from the university without pay for violating the student's privacy. The student failed the analysis part of the qualifying exams twice but appealed the decision, based on a "severe, disabling exam anxiety." The dean of graduate studies agreed, waived the exam requirement, and awarded the degree. Lukacs contends that granting the PhD to this student diminishes the department's reputation and that the dean overstepped his bounds making a decision that rests with the faculty or an appeals committee. Read more details about the case. Lukacs was also in the news recently for winning a case against an airline regarding its estimate of how much baggage is valued at.

Two mathematicians have found a novel technique for measuring the composition of the universe, and it verifies the model of nearly 75% dark energy built by cosmologists. The technique is based on the premise that the formula for calculating the shape of a cosmological object depends on the shape and contents of the universe. Starting with the angles at which different points on an object are observed and the shift in the light captured from the points, one can compute the speed at which the points are moving. From there, one uses a formula--dependent on the universe’s characteristics—to convert the speed to a distance measurement and determine the shape of the object. If one performs this conversion using measurements from an object whose shape is known, one can “test” the accuracy of the formula, and the assumptions about the universe, by comparing the calculated and actual shapes. Known shapes are difficult to find, however, and the mathematicians’ breakthrough was to use a “known” distribution instead: the orientation of pairs of orbiting galaxies.

The authors begin with: "It's very pretty, but what is it? It may look familiar, since we used a similar diagram when discussing Prime Ministers' birthdays. Older readers, and fans of 70's folk art, may recall similar diagrams brought to life with coloured thread and nails hammered into wood. The diagram is known evocatively as a mystic rose. Mathematicians, poets that they are, refer to it as the complete graph on 30 vertices." Polster and Ross go on to explain how the mathematical rose is created. "For the 30-point mystic rose there are 13,800 two-line intersection points, and thousands of multiple intersection points, including 30 seven-line intersections." The authors give away the answer of the number of regions in the 30-point rose, and note how its name may come from its being reminiscent of stained-glass windows in churches, known as rose windows.

The broadcast centers on the recently released report that U.S. ranks 31st of 56 industrialized countries in advanced mathematics skills, and that among the 50 U.S. states, Oregon ranks 8th. Guests on the program were Sarah Schuhl, math instructional coach at Centennial High School in Portland; Dawson Green, a 2004 graduate of Cleveland High School and a part-time math tutor to high school students; Janet S. Hyde, professor of psychology and women's studies at the University of Wisconsin, Madison; Rebecca Goldin, associate professor of mathematics at George Mason University and director of research at STATS (Statistical Assessment Service); and Arthur Benjamin, "mathemagician" and professor of mathematics at Harvey Mudd College in Claremont, CA. The program generated a lot of feedback on the OPB website.

Ping-Tung Chang, professor of mathematics at Matanuska-Susitna College near Anchorage, Alaska, is one of four people named 2010 U.S. Professor of the Year by CASE (Council for Advancement and Support of Education) and The Carnegie Foundation for the Advancement of Teaching. "He uses what he calls the 'grow your own problem-solving approach,' a method developed by the Hungarian mathematician George Pólya in the 1940s to help stimulate their interest in the subject." The Chronicle reports that as a tribute to his inspiration and successful methods, some of his former students have started a scholarship fund in his name that awards at least $500 every year to an in-state student attending either Matanuska-Susitna or its parent institution, the University of Alaska at Anchorage. The U.S. Professors of the Year Program website includes more information and links to Chang's "Passion for Teaching Statement," student introduction, and acceptance speech.

This article provides a breakdown of the key terms for describing a normal distribution–-mean, standard deviation, skew, and kurtosis–-for use by those examining the performance of stocks. Forgoing strict mathematical explanations, the author opts instead for pictorial descriptions of a bell curve to demonstrate these concepts. He then provides fairly simple translations of these parameters into expected fluctuations in stock prices, such as smaller standard deviation equals less risk. A brief description of the contributions of Carl Friedrich Gauss--for whom the Gaussian distribution is named--rounds out this dictionary entry-like article. Those with some background knowledge of these concepts may feel shortchanged by the oversimplified descriptions of the applications, however, while those with little mathematical expertise may be left wondering how standard deviation is computed.

“Before Pythagoras: The Culture of Old Babylonian Mathematics” is a new exhibit at New York University’s Institute for the Study of the Ancient World (ISAW) featuring 13 tablets from 1900-1700 BCE, and is on view until December 17. Alexander Jones, ISAW Professor of History of Exact Sciences in Antiquity, one of the curators of the exhibit along with Christine Proust, ISAW visiting scholar and historian of mathematics and ancient sciences at the Institut Méditerranéen de Recherches Avancées in Marseille, describes the mathematics found in the tablets as “amazing not only in its abundance, but also in its range, from basic arithmetic to really challenging problems and investigations.” The curator also says that it is really valuable that these are actual manuscripts written by Babylonian scribes in cuneiform script, since it gives an insight into how mathematics was taught and used in Old Babylon. These tablets were first transcribed and interpreted by mathematician and historian of science Otto Neugebauer (1899-1990) for two decades, beginning in the 1920s. The exhibit also features some of Neugebauer’s correspondence and manuscripts. In a website devoted to the exhibition, one can see highlights such as a tablet depicting what is widely known as the Pythagorean Theorem (written 1000 years before Pythagoras) and another listing many Pythagorean triples.

The second article is a slide show of 10 of the tablets on The New York Times site, while the third is a review of the exhibition. Image (click for larger version): Old Babylonian tablet containing problems concerning the digging of trenches. Clay, 19th-17th century BCE, Yale Babylonian Collection YBC 4663. Photo by Christine Proust.

About three years ago, while watching his favorite feline Cutta Cutta drinking, MIT engineering professor Roman Stocker became curious about how cats drink without getting their whiskers wet. Using a high-speed camera from the Edgerton Center at MIT to film Cutta Cutta, Stocker found that only the top of the tongue comes into contact with the surface of the liquid before it is quickly withdrawn. Alleyne explains that, “this forms a column of milk between the tongue and the surface, which the cat captures by closing its mouth. This column is created by a balance between gravity pulling the liquid back to the bowl, and inertia.” Determined to understand what was happening, the team built a mechanical model of a cat’s tongue in the lab. They also filmed several other domestic cats—as well as some big cats at a local zoo—lapping. By “slowing down the video they established the speed of the tongue’s movement and the frequency of lapping. Knowing the size and speed of the tongue, they were able to calculate a mathematical formula involving the Froude number—a dimensionless number that characterizes the ratio between gravity and inertia.” In the words of team member Jeffrey Aristoff, a mathematics professor at Princeton University, "the amount of liquid available for the cat to capture each time it closes its mouth depends on the size and speed of the tongue. Our research—the experimental measurements and theoretical predictions—suggests that the cat chooses the speed in order to maximize the amount of liquid ingested per lap. This suggests that cats are smarter than many people think, at least when it comes to hydrodynamics." (A conclusion with which most cats would agree.)

A study by researchers at Stanford University and the University of Munich found that, compared to other countries, the U.S. has very few students with advanced mathematics skills. Only 6 percent of U.S. eighth-graders perform at the advanced level in math, compared with 28 percent of Taiwanese students and more than 20 percent of students in Hong Kong, Finland, and South Korea, the Mercury News reported. Are these results a consequence of the relatively high proportion of immigrants and disadvantaged minority groups in the U.S.? To investigate this question, the study looked at the advanced math skills of two American sub-populations: white students and students from college-educated families. Neither group was outstanding. When compared with all students in other nations, the former group was outperformed by 24 nations and the latter by 16 nations. Read the report, called Teaching Math to the Talented.

In this edition of his “Who’s Counting” column, Paulos describes some of the interesting properties and clever applications of coin flips. In a fair coin flip there are two possible outcomes that are equally likely, namely heads or tails. Since coin flips are independent of each other, the probability of getting a particular sequence of heads or tails is the product of the probabilities for each coin flip (so for example the probability of getting H-T-H-H is ½ x ½ x ½ x ½ =1/16). If the coins are biased in some way, there are tricks for evening out the probabilities. The mathematician John von Neumann devised one such trick to get around biased coins. With his method, the two parties involved in the coin flipping have to flip the coin twice. If it comes up heads both times or tails both times, the coin is to be flipped twice again. If the flip comes up heads, then tails, then the first party wins, if tails then heads the second party wins. Since the flips are independent, it is easy to see that the probabilities of T-H and H-T are the same, and thus we have evened out the probabilities (even if the coin was initially biased towards coming up heads, for example). Paulos explains another method for dealing with unfair coins, and explains how to use von Neumann’s method to obtain other probabilities from coin flips (other than ones with powers of two in the denominator). He then poses a couple of challenge problems for the reader involving sequences of flips.

This issue of Science News had some math at its beginning and some at its end. In Scientific Observations, a quote in each issue, is an excerpt from The Calculus Diaries by science writer Jennifer Ouellette. It begins by noting that basic life forms behave according to rules derived from calculus, and continues

"If worms can do calculus, human beings simply have no excuse for avoiding it. I think scientists have a valid point when they bemoan the fact that it's socially acceptable in our culture to be utterly ignorant of math, whereas it is a shameful thing to be illiterate. We could all be just a little bit mathier... We ought to have some basic understanding of how math in general, and calculus in particular, fits into our cultural framework, and be able to look at a rudimentary equation without breaking into a cold sweat. It is an integral part of our history, after all."

On the last page of the issue is an interview with U.S. Chief Statistician Katherine Wallman about why numbers and understanding them are important. She talks about her job coordinating an office that provides oversight for the government's statistical programs and how with the advent of computers people have more numbers and statistics at their disposal, even though they may not have the expertise to deal with them.

Discovering the solution to any one of the six remaining Millennium problems could earn an individual a million dollar prize from the Clay Mathematics Institute. Even though few members of the general public have the tools to successfully solve these problems, blogger Matt Parker eggs them on with a playful attitude as if to say "What could it hurt to try?" His series of posts kicked off with Goldbach's conjecture as a warm-up to the Millennium problems. He then tackled the Riemann Hypothesis and the P vs NP problem, which one mathematician recently claimed to have solved. The Riemann Hypothesis seems the hardest to make even translucent (much less transparent) to the average science-savvy reader. Parker does a good job of highlighting the striking nature of the pattern that Riemann observed in the primes. But he presents the Zeta function's formula with little explanation. However, just the fact that complex numbers are somehow involved in finding a formula for the number of primes less than a given integer might spark some readers interests.

The posts also give some motivation for answering these questions, with both the Riemann Hypothesis and the P vs. NP problem being related to code-breaking. Even royalty might find non-deterministic polynomial time to be a significant concept; Parker explains why seating the guests at the next royal wedding is an NP problem, and then ponders whether it is in the P category as well. Having a sense of humor is part of Parker's skill set, and on his website he bills himself as a stand-up mathematician and a math communicator. Having studied math and physics before training to be a teacher, he is currently based at Queen Mary College, University of London, and works to recruit young people to the field of mathematics. With that in mind, even if his posts oversimplify the featured problems, maybe they will motivate a young reader or two to sign up for complex analysis or mathematical logic.

American Scientist remembers math-puzzle-maker Martin Gardner in its latest issue. Gardner was an amateur mathematician, a professional magician and a long time Scientific American columnist. His clear presentation, entertaining prose and continous correspondence with readers inspired young and old to partake in solving mathematical problems. The article revisits some of Gardner’s puzzles, which include origami-magic and coin-flipping tricks. Computer scientist Erik D. Demaine also puts forward new puzzles and considers how the use of computers is changing puzzle solving.

During Halloween, kids reach into holes to feel the "brains" (spaghetti), "eyeballs" (peeled grapes), and other pseudo scary substances. Katherine Kuchenbecker, a haptics researcher at University of Pennsylvania, studies ways to trick our sense of touch in a more sophisticated way. Popular Science chose to profile her as one the ten brilliant young scientists to watch this year. Her work might allow better feed back in simulations of surgery and even military battle. Members of Dr. Kuchenbecker's lab record the vibrations created from say, writing on cloth versus paper. Using mathematical modeling these subtle distinctions can be reproduced on demand.

Other young scientists profiled are applying mathematics in fascinating ways. Light bouncing off the shell of a Brazillian beetle triggered material scientist Michael Bartl's sense of curiosity. The color of the beetles scales appeared the same from every angle, which led Dr. Bartl to discover a diamond-shaped photonic crystal that reflects visible light. Now he is working to create a nanostructure that imitates the behavior of the beetles shell.

Observing other animal life provides not only inspiration for inventions but clues as to how our lives might change in the future. Computational biologist Raul Rabadan looks for patterns in large bodies of data collected over time to discover and track unknown viruses. His mathematical tool, Frequency Analysis of Sequence Data, helps researchers look through RNA sequences for evidence of viruses, such as the one that killed large numbers of farmed Atlantic Salmon in Norway or the H1N1 virus found to originate in pigs.

To understand large animal populations, biologist Iain Couzin also employs computational methods and mathematics to model the behavior of herds of animals or colonies of insects as they travel. Even cancer cells exhibit collective behavior that follows some of the same mathematical rules that fish or birds follow as they transition from disorganization to seemingly planned harmony in movement.

While Dr. Couzins studies schools of fish, engineer Maurizio Porfiri schools robots so that they might behave like fish. Dr. Porfini, who is in the Dynamical Systems Laboratory at the Polytechnic Institute of NYU, specializes in underwater robotics, which differs from terrestrial robotics because of challenges in communications and energy use. By following "simple algorithms", the engineer's underwater robots might be able to explore unknown territories without severely disrupting native animals.